Tenant issues

Hi. I am new in the property managwment field and hoping someone can help me figure this out. Our tenant passed away in his apartment last july 5,2015. He paid his july rent on the 1st.
He was the only one in the lease. Now, his son, is staying there and is asking if he can stay until the end of august.
Is there a proper way (legal) to turn over the lease. The son said outright we wont ne able to pay the rent until he has access to his fathers account. So i am worried this might take more than a month.

When is the lease up? Is there a clause that handles breaking it early?

I'd probably let the son stay until the end of August, in hopes of not having to deal with the belongings myself, but I would charge ample late fees for paying late, and maybe insist on weekly progress payments.